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The Phase Diagram of Neutral Quark Matter 41 2.1 Thephasediagramofmasslessquarks The Phase Diagram of Neutral Quark Matter Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften vorgelegt beim Fachbereich Physik der Johann Wolfgang Goethe - Universit¨at in Frankfurt am Main von Stefan Bernhard Ruster¨ arXiv:nucl-th/0612090v1 20 Dec 2006 aus Alzenau in Ufr. Frankfurt 2006 (D 30) 2 vom Fachbereich Physik der Johann Wolfgang Goethe - Universit¨at als Dissertation angenommen. Dekan: Prof. Dr. Aßmus Gutachter: Prof. Dr. Rischke und HD PD Dr. Schaffner-Bielich Datum der Disputation: 14. Dezember 2006 Abstract In this thesis, I study the phase diagram of dense, locally neutral three-flavor quark matter as a function of the strange quark mass, the quark chemical potential, and the temperature, employing a general nine-parameter ansatz for the gap matrix. At zero temperature and small values of the strange quark mass, the ground state of quark matter corresponds to the color–flavor-locked (CFL) phase. At some critical value of the strange quark mass, this is replaced by the recently proposed gapless CFL (gCFL) phase. I also find several other phases, for instance, a metallic CFL (mCFL) phase, a so-called uSC phase where all colors of up quarks are paired, as well as the standard two-flavor color-superconducting (2SC) phase and the gapless 2SC (g2SC) phase. I also study the phase diagram of dense, locally neutral three-flavor quark matter within the framework of a Nambu–Jona-Lasinio (NJL) model. In the analysis, dynamically generated quark masses are taken into account self-consistently. The phase diagram in the plane of temperature and quark chemical potential is presented. The results for two qualitatively different regimes, intermediate and strong diquark coupling strength, are presented. It is shown that the role of gapless phases diminishes with increasing diquark coupling strength. In addition, I study the effect of neutrino trapping on the phase diagram of dense, locally neutral three-flavor quark matter within the same NJL model. The phase diagrams in the plane of temperature and quark chemical potential, as well as in the plane of temperature and lepton- number chemical potential are presented. I show that neutrino trapping favors two-flavor color superconductivity and disfavors the color–flavor-locked phase at intermediate densities of matter. At the same time, the location of the critical line separating the two-flavor color-superconducting phase and the normal phase of quark matter is little affected by the presence of neutrinos. The implications of these results for the evolution of protoneutron stars are briefly discussed. 4 Acknowledgments I am very grateful to my advisor Prof. Dr. Dirk Rischke who suggested the topic for my thesis. He introduced me to quantum field theory and color superconductivity. I learnt a lot in his lectures and in private communication. I thank him for his suggestions and advices. I am very thankful to Prof. Dr. Igor Shovkovy. I thank him for the excellent cooperation, the discussions, suggestions, and advices. I am grateful to our colleagues Verena Werth and PD Dr. Michael Buballa from the Institut f¨ur Kernphysik at the Technische Universit¨at Darmstadt for the teamwork. I thank Hossein Malekzadeh for the cooperation concerning the spin-zero A-phase of color-superconducting quark matter. I am grateful to HD PD Dr. J¨urgen Schaffner-Bielich. I learnt a lot in his lectures, seminars, and in our astro group meetings. I also thank him and Matthias Hempel for the cooperation and discussions concerning the outer crust of nonaccreting cold neutron stars. I am grateful to the computer trouble team for removing computer problems. I am thankful for using the Center for Scientific Computing (CSC) of the Johann Wolfgang Goethe - Universit¨at. I am very grateful to my parents who supported me during the whole time of my study. 6 Contents Abstract 3 Acknowledgments 5 Contents 7 List of Figures 9 List of Tables 11 1 Introduction 13 1.1 The phase diagram of strongly interacting matter . ........ 13 1.2 Colorsuperconductivity .............................. ... 15 1.2.1 The2SCphase .................................. 17 1.2.2 TheCFLphase.................................. 18 1.2.3 Spin-one color superconductivity . .. 20 1.3 Stellar evolution . 21 1.3.1 Theformationofstars.............................. 21 1.3.2 Mainsequencestars ............................... 22 1.3.3 Redgiantsandredsupergiants. 24 1.3.4 Compactstars .................................. 25 1.4 Neutronstars..................................... .. 25 1.4.1 Pulsars ...................................... 27 1.4.2 Structureofneutronstars . ... 28 1.4.3 Propertiesofneutronstarmatter. .... 30 1.4.4 Toymodelsofneutralnormalquarkmatter . .... 33 1.5 Color superconductivity in neutron stars . ....... 36 1.5.1 Toy models of neutral color-superconducting quark matter . ......... 38 2 The phase diagram of neutral quark matter 41 2.1 Thephasediagramofmasslessquarks . ..... 42 2.1.1 Quantumchromodynamics ........................... 42 2.1.2 Theeffectiveactionofquarks . 45 2.1.3 Propagators and self-energies in projector representation ........... 49 2.1.4 The potential part of the effective action of quarks . ...... 49 2.1.5 The kinetic part of the effective action of quarks . .... 51 2.1.6 The pressure of color-superconducting quark matter . ........ 54 2.1.7 Resultsatzerotemperature. .. 58 2.1.8 Resultsatnonzerotemperature. ... 62 2.2 The phase diagram with a self-consistent treatment of quark masses ........ 66 2.2.1 Model....................................... 66 2.2.2 Results ...................................... 75 2.3 The phase diagram with the effect of neutrino trapping . ....... 83 8 CONTENTS 2.3.1 Model....................................... 84 2.3.2 Simplified considerations . 86 2.3.3 Results ...................................... 89 3 Conclusions 101 3.1 Summaryanddiscussion............................... 101 3.2 Openquestionsandoutlook.. .. .. .. .. .. .. .. .. .. .. .. 104 A Definitions of matrices 109 A.1 ThePaulimatrices................................... 109 A.1.1 Spinprojectors .................................. 109 A.2 MatricesinDiracspace ................................ 110 A.2.1 ProjectorsinDiracspace ............................ 110 A.3 The generators of the SU (3)group........................... 111 B Useful formulae 113 B.1 Non-interacting massless fermions and antifermions at nonzero temperature . 113 B.2 TheinverseDiracpropagator . .. 114 B.3 Thetree-levelquarkpropagator. ....... 114 B.4 TheFeynmangaugedgluonpropagator . ..... 115 B.5 The determinant of the inverse quark propagator . ......... 116 B.6 Thelogarithmofthedeterminant. .. 118 B.7 TheDiractrace..................................... 118 B.8 Thetraceofthelogarithm............................. 119 B.9 Cubicequations..................................... 120 B.10 Summation over the fermionic Matsubara frequencies . .......... 123 C Zusammenfassung 127 Bibliography 133 Lebenslauf 141 List of Figures 1.1 The knowledge about the phase diagram of strongly interacting matterin2003. 14 1.2 The one-gluon exchange interaction between two quarks in QCD. .......... 16 1.3 TheHertzsprung-Russelldiagram. ...... 22 1.4 Theproton-protoncycles. ..... 23 1.5 Thecarbon-nitrogen-oxygencycles. ........ 23 1.6 Theevolutionofneutronstars. ..... 26 1.7 Aschematicrepresentationofapulsar.. ....... 28 1.8 Cross-sectionsofneutronstars. ........ 29 1.9 The quark-hadron phase transition in neutron stars. .......... 30 1.10 The low-energy part of the dispersion relations. ......... 37 2.1 The sunset-type diagram, and the double-bubble diagram. .......... 46 2 2.2 The gap parameters as a function of ms/µ. ...................... 58 2 2.3 The chemical potentials of electric and color charge as a function of ms/µ...... 59 2.4 The gap parameters as a function of the quark chemical potential. 59 2.5 The quark number densities as a function of the quark chemical potential. 60 2.6 The quasiparticle dispersion relations. ..... 61 2.7 The temperature dependence of the gaps for a small strange quarkmass. 62 2.8 The near-critical temperature dependence of the gaps for a small ms......... 63 2.9 The temperature dependence of the gaps for a large strange quarkmass.. 63 2.10 The temperature dependence of the electrical and color chemical potentials. 64 2.11 The phase diagram of massless neutral three-flavor quark matter........... 65 3 2.12 The phase diagram of neutral quark matter without neutrinos for GD = 4 GS .... 76 2.13 The phase diagram of neutral quark matter without neutrinos for GD = GS . ... 76 3 2.14 Quark masses, gap parameters, and chemical potentials at GD = 4 GS ........ 77 2.15 Quark masses, gap parameters, and chemical potentials at GD = GS ......... 79 3 2.16 The number densities of quarks and electrons at T = 0 and GD = 4 GS. ...... 80 2.17 The number densities of quarks, electrons, and muons at T = 0 and GD = GS. .. 81 2.18 The pressures of different phases of neutral color-superconducting quark matter. 82 2.19 Ratio of down and up quark chemical potentials as a function of µLe /µ....... 87 2.20 The three-dimensional phase diagram of neutral three-flavorquarkmatter.. 90 2.21 The phase diagram of neutral quark matter at µLe =200MeV. ........... 91 2.22 The phase diagram of neutral quark matter at µLe =400MeV. ........... 91 2.23 Quark masses, gap parameters, and chemical potentials at µLe =200MeV. 92 2.24 Quark masses, gap parameters, and chemical potentials at µLe =400MeV. 93 2.25 The dependence of the electron family lepton fraction YLe on µ............ 94 2.26 Thephasediagramoftheouterstellarcore.
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